Asymptotics of the generating function for the volume of the Wiener sausage (Q1333575)

We consider the generating function \(\mathbb{E} \exp (\lambda | C_ \varepsilon (t) |)\) of the volume of the Wiener sausage \(C_ \varepsilon (t)\), which is the \(\varepsilon\)-neighbourhood of the Wiener path in the time interval \([0,t]\). For \(\lambda < 0\), the limiting behaviour for \(t \to \infty\), up to logarithmic equivalence, had been determined in a celebrated work of Donsker and Varadhan. For \(\lambda > 0\) it had been investigated by the first author and \textit{B. Tóth} [ibid. 88, No. 2, 249-259 (1991; Zbl 0717.60093)], but in contrast to the case \(\lambda < 0\), there is no simple expression for the exponential rate known. We determine the asymptotic behaviour of this rate for small and large \(\lambda\).